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We present a novel approach for analytically reducing a family oftime-dependent multi-state quantum control problems to two-state systems. Thepresented method translates between $SU(2)XSU(2)$ controlled $n^{2}$-statesystems and two-state systems, such that the former undergo complete populationtransfer (CPT) if and only if the latter reach specific states. For even n, themethod translates any two-state CPT scheme to CPT schemes in $n^{2}$-statesystems. In particular, facilitating CPT in a four-state system via realtime-dependent nearest-neighbors couplings is reduced to facilitating CPT in atwo-level system. Furthermore, we show that the method can be used for operatorcontrol, and provide conditions for producing several universal gates forquantum computation as an example. In addition, we indicate a basis forutilizing the method in optimal control problems.

A quantum retrograde canon: Complete population transfer in $n^{2}$-state systems